cohomology of noncompact surfaces

This paper is motivated by the question of whether nonzero L2-harmonic differentials exist on coverings of a Riemann surface of genus > 2. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of L2-homology and the intersection number of LU-cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of L2-harmonic one-cycles and the transience of random walks on a planar surface is shown.